HUE UNIVERSITY
COLLEGE OF EDUCATION
-oOo-
LE HOANG MAI
ON JACOBSON RADICAL,
JS -RADICAL AND
RADICAL CLASSES OF
HEMIRINGS
Speciality: Algebra and Number Theory
Code: 62 46 01 04
SUMMARY OF
DOCTORAL DISSERTATION IN MATHEMATICS
HUE - 2016
The work was completed at: Faculty of Mathematics, College of Education, Hue University.
Supervisor: Assoc. Prof. DrSc. Nguyen Xuan Tuyen.
First referee:...................................................................................................
Second referee:...............................................................................................
Third referee:.................................................................................................
The dissertation will be defended at the Council level dissertation
dots Hue University in the meeting:............ ....................................................
In ........... hour ............. day............. month ............ year ...........
Can learn dissertation at the library:.........................................................
1
PREAMBLE
1 Why choose topics
The radical concept was first studied by Cartan for finite dimension Lie
algebras over algebra closed fields. The radical of a finite dimension Lie algebra
A is a maximal solvable ideal of A and it is the sum of all solvable ideals of A.
A Lie algebra A is called semisimple if its radical is zero. Cartan showed that a
semisimple Lie algebra is a finite direct sum of simple Lie algebras. Moreover,
he also described the finite dimension simple Lie algebras over algebra closed
fields. Therefore, structure of the finite dimension semisimple Lie algebra is fully
determined.
Wedderburn has expanded the above result for finite dimension associative
algebras over fields. He defined the such radical of this algebra A, denoted rad(A),
that is a maximal nilpotent ideal of A and it is also the sum of all nilpotent ideals
of A. Similarly Cartan, Wedderburn called that a finite dimension algebra A is
semisimple if rad(A) = 0. He has proven that the finite dimension algebra A is
semisimple if and only if it is a finite direct sum of the finite dimension simple
algebras Ai , where each Ai is a matrix algebra over a finite dimension division
algebra.
Artin expanded Wedderburn’s Theorem for rings satisfied the minimal condition (called an artinian ring). With such a ring R, the sum of all nilpotent
ideals in R is nilpotent, therefore R has a maximal nilpotent ideal rad(R), called
Wedderburn radical of R. So, Wedderburn’s Theorem for semisimple and simple
algebras has been successfully extended to the one-sided artinian rings. However,
for rings R that are not one-sided Artin, the sum of all nilpotent ideals in R is
not nilpotent and so, R does not have a maximal nilpotent ideal, therefore we
have no a concept of radical for any rings.
In 1945, Jacobson [25] defined a concept of radical (called Jacobson radical )
for any associative ring, it is the sum of all right quasiregular right ideals. In particular, if R is the one-sided artinian ring then Jacobson radical and Wedderburn
radical concepts of R coincide. From now on, the Jacobson radical became one
of the useful tools to study the structure of the rings. The Jacobson radical of
ring theory and related problems were presented completely and systematically
in references such as: Gardner-Wiegandt [11], Lam [36] and Anderson-Fuller [6].
The concept of a hemiring was introduced by Vandiver [56] in 1934, is a
2
generalized concept of an associative ring, with the sense that does not require
symmetry of the additive operation. In the 30s of the 20th century, a concept
of a hemiring has not been much interested for mathematical community. The
importance of hemirings in theoretical computer science was first recognized by
Schützenberger [52]. Today, hemirings are developed both in terms of theory and
application. The properties and applications of hemirings and related problems
were presented in references such as: Golan [13], Berstel-Reutenauer [8] and
Polák [18].
Recently, an additively idempotent hemiring (also called an idempotent
hemiring by some authors) is interested by mathematicians such as: Gathmann
[12] and Izhakian-Rowen [23] since additively idempotent hemirings are the
heart of the relatively new subject of tropical geometry and tropical algebra.
With that, a concept of a simple left semimodule over an additively idempotent
hemiring is also interested in research as: Izhakian-Rhodes-Steinberg [24] describe all simple left semimodules over an idempotent finite semigroup algebra
BS (S is a finite semigroup), Kendziorra-Zumbrägel [32] show that there exists a simple left semimodule over an additively idempotent finite semiring and
Katsov-Nam-Zumbrägel [29] also show that there exists a simple left semimodule over a complete hemiring R such that R has only trivial congruences with
RR 6= 0. However, the existence of a simple left semimodule over any hemiring
is an unsolved problem.
From these problems suggest us to study the structure of hemirings. Similar
to rings, in this dissertation we use one of the useful tools to study the structure
of hemirings that is the radical tool. Generally, the radical of a hemiring R
includes all bad elements of R such that the factor hemiring of R over its radical
does not contain bad elements.
The radical of a hemiring started being considered by mathematicians since
the 50s of the 20th century. Specially, in 1951, Bourne [9] defined Jacobson radical (J -radical) of a hemiring based on the one-side semiregular ideals. Moreover,
Bourne has also proved that every nilpotent left (right) ideal of a hemiring is contained in J -radical [9, Theorem 7] and calculated J -radical of a matrix semiring
over a semiring [9, Theorem 9]. In 1958, Bounne and Zassenhaus [10] has introduced a such special ideal class of a hemiring whose ideal is called subtractive
(or k -ideal ) and proved J -radical of a hemiring is also a subtractive ideal.
The Jacobson radical of hemirings continues to study by the representation
theory. In [21], Iizuka used irreducible left semimodules for characterization of J -
3
radical of hemirings [21, Theorem 8]. He defined also strong primitive ideals of a
hemiring and characterization of J -radical of a hemiring R being the intersection
of all strong primitive ideals of R [21, Theorem 6], and showed relationship
between J -radical of a hemiring R and Jacobson radical of a ring of differences
D(R) [21, p. 420]. Moreover, he introduced a special ideal class, these ideals are
called h-ideals, of hemirings and proved J -radical of a hemiring is also a h-ideal.
In [38], LaTorre proved J -radical of a hemiring is a right k -ideal (h-ideal)
generated by the set of all right semiregular right k -ideals (h-ideals) [38, Theorem 3.1] and if R is a ring then two concepts of Jacobson radical of a ring
and Jacobson radical of a hemiring coincide [38, Theorem 3.2]. Moreover, he
established some familiar properties related to Jacobson radical in ring theory
for hemirings. Specially, LaTorre described the structure of J -semisimple additively regular hemirings [38, Theorem 3.4]. However, results related to J -radical
of hemirings in the current time is very limited more than results related to
Jacobson radical in ring theory.
Recently, Katsov-Nam received some results related to J -radical of hemirings
[26, Section 3 and 4], especially the results on the structure of hemirings adopting J -radical, such as Hopkins theorem for an artinian hemiring [26, Corollary
4.4] and a theorem on the structure for a primitive hemiring [26, Theorem 4.5].
However, J -radical being limited by an additively idempotent hemiring belonged
to its induced radical class, i.e., if R is an additively idempotent hemiring then
J(R) = R ([26, Example 3.7] or [53, Proposition 2.5 ]). To remedy this problem,
Katsov-Nam introduced a concept of Js -radical (generalized from Jacobson radical in ring theory) for hemirings by using simple left semimodules [26, p. 5076]
and received the structure description theorem of Js -semisimple additively idempotent finite hemirings thought this radical [26, Theorem 3.11]. Simultaneously,
they showed that J -radical and Js -radical coincide for rings but in the general
case of hemirings are different, for example, as additively idempotent hemirings
[26, Example 3.7], and pointed out the relationship between them for additively
regular hemirings and commutative hemirings [26, Proposition 4.8]. However,
relationship between J -radical and Js -radical of hemirings in the general case
is unknown. In light of these observations, a natural problem raised to consider
the relationship between these radicals.
Problem [26, Problem 1] Describe the subclass of all hemirings R such that
Js (R) ⊆ J(R), particularly, with Js (R) = J(R).
In this dissertation, we continue to use the tools of J -radical and Js -radical
4
to study the structure of hemirings, establish some important results related
to Jacobson radical in ring theory for hemirings, describe relationship between
J -radical and Js -radical for some classes of hemirings, thereby, we partially answered [26, Problem 1].
Moreover, this dissertation also interested Kurosh-Amitsur radical of hemirings. Early 50s of the 20th century, Amitsur [2, 3, 4] and Kurosh [34] are the first
mathematicians independently discovered that all classical radicals always have
certain common properties and they used these algebra properties to axiomatize
a defination of an abstract radical class. In 1988, Kurosh-Amitsur radical for a
common algebra categories is proposed by Márki-Mlitz-Wiegandt [46]. In 2004,
Kurosh-Amitsur radical of rings and related results were presented in a systematic way by Gardner-Wiegandt [11]. There, with each radical class γ of rings one
always defined a radical operator or radical calculation (is called γ -radical or
Kurosh-Amitsur radical of rings) and on the contrary, for each radical operator
ρ one always defined a radical class of rings.
In 1983, Olson-Jenkins [48] has generalized a concept of a radical class in
ring theory to hemirings and then some related problems to the radical class
of hemirings are presented in a series of works [49, 50, 51] by Olson and his
colleagues. Moreover, Kurosh-Amitsur radical of semifield categories is studied
by Weinert-Wiegandt [59, 60, 61], of group categories by Krempa-Malinawska
[33] and Li-Zhang [42].
Recently, Kurosh-Amitsur radical of hemirings continues to be studied. In
[15, p. 652], Hebisch-Weinert has built radical classes from weakly special and
special classes. Morak [47] has built three concepts of Kurosh-Amitsur radical of
hemirings independently, that is a radical class, a semisimple class and a radical
operator and Hebisch-Weinert [17, Theorem 3.6] showed corresponding one-one
among the three concepts. In [16, Theorem 3.4], Hiebsch-Weinert proved that
from a radical class of rings one can always built a radical class of hemirings.
Moveover, Morak [47, Theorem 5.3] also built a radical class from a regular class
of hemirings which is called upper radical class.
In [11, p. 28], the lower radical class of a class δ of rings is intersection of all
radical classes contain δ and it is the minimal radical class contained δ , denoted
by Lδ . There are several construction methods of the lower radical class of a
class δ of rings as method of Watters [58], method of Kurosh [34] and method
of Lee [40]. The lower radical class of a class of hemirings is defined similar
in ring theory and the lower radical class of a class A of hemirings is denoted
5
also by LA. In [63, Theorem 2.6], Zulfiqar has built a lower radical class of
a class of hemirings by similar method of Watters. Moreover, Zulfiqar [62, 64]
also generalized concepts of the sum of two radical classes and the intersection
of a radical class with the sum of two radical classes, constructed by Lee and
Propes [11] in ring theory, for hemirings. Hereditary properties of a radical class
of rings is studied by Anderson-Divinsky-Sulínski [5] and Morak [47, Section 6]
generalized these properties for radical class of hemirings.
However, results related to Kurosh-Amitsur radical of hemirings until the
present time are still more modest than corresponding results of Kurosh-Amitsur
radical of rings.
With the above reasons, we choose the theme “On Jacobson radical, Js radical and radical classes of hemirings” for my dissertation. The following
issues were the subject of our research focus.
(1) Using tools of J -radical and Js -radical to study the structure of some
hemirings and establish some important results related to Jacobson radical in
ring theory for hemirings.
(2) Establish relationship between J -radical and Js -radical for some classes of
hemirings (these results partly answer [26, Problem 1]). Describe some classes of
hemirings which are Js -semisimple left (right) V-semirings (these results partly
answer [1, Problem 1]).
(3) Look at the issues related to a radical class of hemirings, such as propose
the concept of an accessible subhemiring and give a characterization of a radical
class by concepts of accessible subhemiring and homomorphism, build a radical
class from a class of hemirings and research hereditary properties of a radical
class of hemirings.
2 Research purpose
We give a complete description of J -semisimple or Js -semisimple hemirings and establish some important results related to Jacobson radical in ring
theory for hemirings. Compare Js -radical and Nil radical for zerosumfree commutative semirings. Establish a necessary and sufficient condition for J -radical
and Js -radical such that they coincide over semisimple semirings, additively π regular semirings, antibounded semirings and V-semirings. Describe some semirings which are Js -semisimple left (right) V-semirings. Characterize a radical class
of hemirings by accessible subhemirings, build a lower radical class of a class of
hemirings and establish a necessary and sufficient condition such that an upper
radical class of a regular class of hemirings is hereditary.
6
3 Object and scope of studying
3.1 Research subjects:
- J -radical, Js -radical of hemirings.
- Radical class of hemirings.
3.2 Research scope:
Associative algebra. Semiring and semimodule theory.
4 Research Methods
- Research method of pure mathematics and specific method of semiring and
semimodule theory.
- Use radicals such as: J -radical, Js -radical and radical class to study the
structure of hemirings and relative problems.
5 Scientific and practical meaning
Describe the structure of J -semisimple additively π -regular semirings. Prove
that there exists a simple left semimodule over an additively idempotent semiring, and Js -radical equal Nil radical for zerosumfree commutative semirings,
establish a similar result of Snapper on Jacobson radical of a polynomial ring
in ring theory for zerosumfree commutative semirings and we give a complete
description of Js -semisimple zerosumfree commutative semirings. Answer partly
Problems [26, Problem 1] and [1, Problem 1]. Characterize some radical classes
of hemirings by accessible subhemirings and homomorphisms. Build the lower
radical class of a class of hemirings, a class of homomorphically closed hemirings. Establish a necessary and sufficient condition for the upper radical class of
a regular class of hemirings is hereditary. Prove that the upper radical class of
a regular class of semirings is hereditary.
6 Overview and structure of the dissertation
7
Chapter 1
PREPARATIVE KNOWLEDGE ON
HEMIRINGS AND SEMIMODULES
In this chapter, we use references [9], [13], [21] and [26] to present some concepts and properties related to hemirings and semimodules. These are necessary
to present the main chapters of the dissertation (Chapter 2 and Chapter 3). This
chapter consists of four subsections: Hemirings and semimodules; Congruence relations, factor hemirings and factor semimodules; Hemiring homomorphisms and
semimodule homomorphisms; Conclusion of Chapter 1.
1.1 Hemirings and semimodules
In this subsection, we present some concepts and examples related to hemirings and semimodules such as: hemiring, semiring, subhemiring, ideal, semimodule, subsemimodule,...
1.2 Congruence relations, the factor hemirings and the
factor semimodules
Here, we recall congruences over hemirings and semimodules, build factor
hemirings and factor semimodules. Moreover, we give some examples and remarks on these concepts.
1.3 Hemiring homomorphisms and semimodule homomorphisms
In this subsection, we recall some concepts and results related to hemiring
homomorphisms and semimodule homomorphisms. We give examples and remarks to show different of hemiring and semimodule homomorphisms with ring
and module homomorphisms.
1.4 Conclusion of Chapter 1
8
Chapter 2
ON JACOBSON RADICAL, JS -RADICAL
OF HEMIRINGS
In this chapter, we use Jacobson radical (J -radical) and Js -radical to describe
structure of additively π -regular semirings, zerosumfree commutative semirings.
Specially, we study relationship between Jacobson radical and Js -radical of
semisimple semirings, additively π -regular semirings, antibounded left artinian
semirings and left artinian left V-semirngs, and relationship between Js -radical
and Nil radical of zerosumfree commutative semirings. Describe some classes of
semirings which are Js -semisimple left (right) V-semirings. Establish some similar results of Hopkins for nilpotent Jacobson radical and of Snapper for Jacobson
radical of a polynomial ring in ring theory for semirings. This chapter consists
of five subsections: On Jacobson radical of hemirings; On Js -radical of hemirings; On relationship between Jacobson radical and Js -radical of hemirings; On
Js -semisimple left (right) V-semirings and Conclusion of Chapter 2. The main
results of this chapter are taken from results in papers [53], [43], [44] and [45].
2.1 On Jacobson radical of hemirings
In 1951, Bourne [9] used the left (right) semiregular ideal to define Jacobson
radical of a hemiring.
Definition 2.1.1. ([9, Definition 3]) Let R be a hemiring and I be a right
ideal of R. The right ideal I is called right semiregular, if for every pair of elements
i1 , i2 ∈ I there exist elements j1 , j2 ∈ I such that:
i1 + j1 + i1 j1 + i2 j2 = i2 + j2 + i1 j2 + i2 j1 .
A left semiregular ideal of a hemiring is defined similarly. The ideal of a hemiring
is called semiregular if it is both left and right semiregular.
Definition 2.1.4. ([9, Definition 4 and Theorem 4]) Let R be a hemiring.
(1) The sum of all the right semiregular ideals of R, denoted J(R), is called
Jacobson radical (J -radical ) of a hemiring R.
(2) A hemiring R is called J -semisimple if J(R) = 0.
9
Example 2.1.5. (1) If R is a ring then J -radical J(R) coincide with Jacobson
radical in ring theory. Indeed, according to [25, Definition 2], Jacobson radical
of a ring R is the sum of all right quasiregular right ideals of R. Therefore,
according to Remark 2.1.2(2), J -radical J(R) coincide with Jacobson radical of
ring R.
(2) If R is an additively idempotent hemiring then J(R) = R. Indeed, according to Remark 2.1.2(3), R is a right semiregular ideal of R. Therefore, from
Definition 2.1.4, we have J(R) = R.
(3) We always have J(N) = 0. Indeed, according to Remark 2.1.2(4), N has
only the zero ideal which is a right semiregular ideal. Therefore, from Definition
2.1.4, we have J(N) = 0.
(4) If R is a hemiring and Mn (R) (n ≥ 1) is a matrix hemiring over R, then
J(Mn (R)) = Mn (J(R)) [26, Theorem 5.8(iii)].
Iizuka [21] used irreducible left semimodules to characterization of Jacobson
radical of hemirings.
Definition 2.1.6. ([21, Definition 5]) For a hemiring R, a cancellative left
R-semimodule M 6= 0 is called irreducible if and only if for an arbitrarily fixed
pair of u1 , u2 ∈ M with u1 6= u2 and any x ∈ M , there exist a1 , a2 ∈ R such that
x + a1 u1 + a2 u2 = a1 u2 + a2 u1 .
Theorem 2.1.8. ([21, Theorem 8]) Let R be a hemiring. Then,
J(R) = ∩{(0 : M )R | M ∈ J },
where J is the set of all irreducible left R-semimodules. If J = ∅ then for convention ∩{(0 : M )R | M ∈ J } equal R.
Using Theorem 2.1.10 and Theorem 2.1.11, we give a complete description
of J -semisimple additively π -regular semirings. This result is an extension of
Latorre [38, Theorem 3.4]. First, we recall a concept of additively π -regular
semirings.
Definition 2.1.12. ([14] or [19, p. 1496]) A semiring R is called additively
π -regular if for any x ∈ R, there exist a natural number n and an element y ∈ R
such that nx + y + nx = nx.
Theorem 2.1.14. For an additively π -regular semiring R, the following conditions are equivalent:
10
(1) R is a J -semisimple semiring;
(2) R is a J -semisimple ring;
(3) R is a ring isomorphic to a subdirect product of primitive rings.
According to Remark 2.1.13(3), if R is a finite semiring then R is an additively π -regular semiring. Therefore, from Theorem 2.1.14 and Theorem of
Wedderburn-Artin in ring theory [36], we immediately obtain the following result.
Corollary 2.1.15. A finite semiring R is J -semisimple if and only if
R∼
= Mn1 (F1 ) × Mn2 (F2 ) × ... × Mnk (Fk ),
where F1 , ..., Fk are finite fields and n1 , ..., nk are positive integers.
Next, we prove a lemma which is necessary to prove the following results of
dissertation.
Lemma 2.1.16. For hemirings R and S , then,
(1) J(R ⊕ S) = J(R) ⊕ J(S);
(2) If R is a division semiring then J(R) = Z(R).
Theorem of Hopkins [36, Theorem 4.12] for a nilpotent Jacobson radical in
ring theory is stated as follows: Let R be a left artinian ring with unit. Then,
Jacobson radical J(R) is a nilpotent maximal left ideal and it is also a nilpotent
maximal right ideal. However, this statement is not true for general semirings.
For example, Boolean field B is a left artinian semiring and its Jacobson radical
J(B) = B is not nilpotent. We conclude this subsection by establishing a similar
result of Hopkins theorem for additively cancellative hemirings.
Lemma 2.1.17. Let R be an additively cancellative hemiring. If right Rsemimodule R2 is artinian then the right R-semimodule D(R) is also artinian.
Theorem 2.1.18. Let R be an additively cancellative hemiring such that R2
is an artinian right R-semimodule. Then, the Jacobson radical J(R) is nilpotent
and R satisfies the condition ACC for subtractive right ideals.
2.2 On Js -radical of hemirings
First, we recall a concept of a simple left semimodule over a hemiring. This
concept has been studied by several authors in recent times, such as: Zumbrägel
11
[65], Izhakian-Rhodes-Steinberg [24], Kendziorra-Zumbrägel [32], Katsov-Nam
[26], Katsov-Nam-Zumbrägel [29], Kepka-Němec [30] and Kepka-KortelainenNěmec [31].
Definition 2.2.1. ([65, Definition 3.7] or [26, p. 5074]) For a hemiring R, a
left R-semimodule M is called simple if the following conditions satisfy:
(1) RM 6= 0;
(2) M is minimal;
(3) M has only trivial congruences.
Example 2.2.3. (1) From Remark 2.2.2(2), if R is a ring then concepts of
simple left R-semimodule and simple left R-module coincide.
(2) Let R be a zerosumfree entire hemiring. Then, B is a simple left Rsemimodule. Indeed, the mapping f : R −→ B defined by f (0) = 0 and f (x) = 1
for every 0 6= x ∈ R, is a hemiring semiisomorphism. For B is a simple left Bsemimodule, so B is also a simple left R-semimodule with scalar multiplication
defined by
rb := f (r)b, for every r ∈ R, b ∈ B.
(3) Let (M, +, 0) be a commutative monoid and End(M ) be a semiring of
endomorphisms (see Example 1.1.2(6)). Then, M is a left End(M )-semimodule
with scalar multiplication defined by
f m := f (m), for every m ∈ M , f ∈ End(M ).
According to [29, Proposition 4.2], if (M, +, 0) is a nonzero idempotent commutative monoid then M is a simple left End(M )-semimodule.
From Remark 2.2.2(2), there exists a minimal left semimodule but not simple. However, the following proposition indicates that we can create a simple left
semimodule from a minimal left semimodule.
Proposition 2.2.4. Let R be a hemiring and M be a minimal left Rsemimodule. Then, there exists a maximal congruence ρ over M such that M :=
M/ρ is a simple left R-semimodule.
The concept of simple left semimodules over a hemiring has been studied
by several mathematicians. In 2011, Izhakian-Rhodes-Steinberg have described
12
all simple left semimodules over an idempotent finite semigroup algebra BS (S
is a finite semigroup) [24, Theorem 4.4]. In 2013, Kendziorra-Zumbrägel show
that there exists a simple left semimodule over an additively idempotent finite
semiring [32, Proposition 2.17]. In 2014, Katsov-Nam-Zumbrägel show that there
exists a simple left semimodule over a complete hemiring which has only trivial
congruences and RR 6= 0 [29, Proposition 4.5]. Next, we prove that there exists
a simple left semimodule over an additively idempotent semiring.
Theorem 2.2.5. For an additively idempotent semiring R, there exists a
simple left R-semimodule.
In 2014, Katsov-Nam [26] use simple left R-semimodules to define Js -radical
of a hemiring R.
Definition 2.2.7. ([26, p. 5076]) Let R be a hemiring.
(1) The subtractive ideal Js (R) := ∩{(0 : M )R | M ∈ J 0 }, where J 0 is the set
of all simple left R-semimodules, is called Js -radical of a hemiring R. If J 0 = ∅
then for convention Js (R) = R.
(2) A hemiring R is called Js -semisimple if Js (R) = 0.
According to [21, Theorem 2], J(I) = I ∩ J(R) for every ideal I of a hemiring
R. Next, we prove a similar result for Js -radical.
Proposition 2.2.10. Let I be an ideal of a hemiring R. Then,
Js (I) = I ∩ Js (R).
Next, we describe Js -radical of a zerosumfree commutative semiring by its
nilpotent elements.
Recall that Nil radical of a semiring R is intersection of all the prime ideals
of R, denoted by N il(R) [13, p. 91]. Symbols P r(R) and P rm (R) are the sets of all
prime ideals and minimal prime ideals, respectively, of a semiring R. According
to Proposition 1.1.11 and [13, Proposition 7.28], if R is a commutative semiring
then N il(R) = ∩P ∈P r(R) P = ∩P ∈P rm (R) P = {r ∈ R | ∃ n ≥ 1 : rn = 0}.
Proposition 2.2.11. If R is a commutative semiring then
N il(R) ⊆ Js (R).
Theorem 2.2.12. Let R be a zerosumfree commutative semiring. Then,
Js (R) = N il(R).
13
In ring theory, Snapper calculates Jacobson radical of a polynomial ring
over a commutative ring with unit [36, Theorem 5.1] as follows: Let R be a
commutative ring with unit and R[x] be a polynomial ring over R. Then,
J(R[x]) = N il(R[x]) = N il(R)[x].
Using Theorem 2.2.12, we establish a similar result of Snapper for a polynomial semiring over a zerosumfree commutative semiring.
Corollary 2.2.15. Let R be a zerosumfree commutative semiring and R[x]
be a polynomial semiring over R. Then, Js (R[x]) = N il(R[x]) = N il(R)[x].
Katsov-Nam give a complete description of Js -semisimple finite additively
idempotent hemirings [26, Theorem 3.11]. Using Theorem 2.2.12, we give a complete description of Js -semisimple zerosumfree commutative semirings.
Corollary 2.2.18. Let R be a zerosumfree commutative semiring. Then, the
following conditions are equivalent:
(1) R is Js -semisimple;
(2) R is quasi-positive;
(3) R is semiisomorphic to a subdirect product of its maximal entire quotients.
2.3 On relationship between Jacobson radical and Js radical of hemirings
In this subsection, we establish a necessary and sufficient condition under
which J -radical and Js -radical coincide over some classes of semirings. These
results partly answer [26, Problem 1].
• First, we answer over semisimple semirings.
Definition 2.3.1. ([27, p. 417]) A semiring R is called left semisimple if a
left semimodule R R is a direct sum of minimal left ideals of R.
Theorem 2.3.4. Let R be a left semisimple semiring. Then, Js (R) = J(R)
if and only if Z(R) = 0.
• Over additively π -regular semirings. A additively π -regular semiring was
mentioned in Definition 2.1.12. The following proposition is an extension [26,
Proposition 4.8] of Katsov-Nam.
14
Proposition 2.3.5. If R is an additively π -regular semiring then Js (R) ⊆
J(R).
Theorem 2.3.6. Let R be an additively π -regular semiring. Then, Js (R) =
J(R) if and only if R is a ring.
• Over antibounded semirings. For a semiring R, set
P (R) := V (R) ∪ {1 + r | r ∈ R}.
It is easy to see that P (R) is a subsemiring of R.
Definition 2.3.7. ([1, p. 4637]) A semiring R is called antibounded if P (R) =
R.
Theorem 2.3.10. If R is an antibounded semiring then Js (R) ⊆ J(R).
Theorem 2.3.11. Let R be a left artinian antibounded semiring. Then,
Js (R) = J(R) if and only if R is a left artinian ring.
• And over left V-semirings.
Definition 2.3.12. ([20, p. 222]) (1) A semiring R is called a left (right) Vsemiring if every left (right) R-semimodule, which only has trivial congruences,
is injective.
(2) A left R-semimodule M is called an essential extension of a R-subsemimodule
L if for every semimodule homomorphism γ : M −→ N of left R-semimodules
then semimodule homomorphisms γi and γ are simultaneously injective, where
i : L M is an embedding.
Proposition 2.3.14. If R is a left V-semiring then Js (R) ⊆ J(R).
Theorem 2.3.16. Let R be a left artinian left V-semiring. Then, Js (R) =
J(R) if and only if R is a left V-ring.
2.4 On Js -semisimple left (right) V-semirings
In this subsection, we describe some classes of semirings which are Js -semisimple
left (right) V-semirings. These results partly answer [1, Problem 1].
As was shown in [35, Theorem 3.75], all left (right) V-rings R are J semisimple rings, i.e., J(R) = 0. However, this is not true for left (right) Vsemirings, in general. For example, the Boolean semifield B is a left (right) Vsemiring and J(B) = B. Moreover, [1, Theorem 3.14] shows that: A left (right)
15
V-semiring R is J -semisimple if and only if R is a left (right) V-ring. However,
it is easy to see that a proper division semiring is a Js -semisimple left (right)
V-semiring, and hence, an analogy of [1, Theorem 3.14] for the Js -semisimple
left (right) V-semirings is not true. In light of this, a problem raised [1, Problem
1] as follows: Describe all Js -semisimple left (right) V-semirings.
First, we give a complete description of such semisimple semirings that are
Js -semisimple left (right) V-semirings.
Theorem 2.4.1. For a simple semiring R, the following conditions are equivalent:
(1) R is a Js -semisimple left (right) V-semiring;
(2) R ∼
= Mn1 (D1 ) × . . . × Mnr (Dr ), where D1 , . . . , Dr are either division rings
or zeroic proper division semirings and n1 , n2 , ..., nr are positive integers.
Next, we consider over congruence-simple semirings, simple semirings.
Theorem 2.4.5. Let R be a semiring. If one of the following is satisfied
then R is a Js -semisimple left (right) V-semiring.
(1) R is a simple semiring with a infinite element;
(2) R is a congruence-simple left (right) artinian, left (right) subtractive
semiring.
Finally, we consider over zeroic antibounded semirings.
Theorem 2.4.7. If R is a zeroic antibounded semiring then R is a Js semisimple left (right) V-semiring.
2.5 Conclusion of Chapter 2
(1) We give a complete description of J -semisimple additively π -regular
semirings (Theorem 2.1.14). Moreover, prove a similar result of Hopkins on
nilpotent Jacobson radical in ring theory for additively cancellative hemirings
(Theorem 2.1.18).
(2) We prove that there exists a simple left semimodule over an additively
idempotent hemiring (Theorem 2.2.5), and prove that Js -radical and Nil radical coincide over zerosumfree commutative semirings (Theorem 2.2.12). From
this result, we obtained a similar result of Snapper on Jacobson radical of polynomial rings in ring theory for zerosumfree commutative semirings (Corollary
16
2.2.15), and give a complete description of Js -semisimple zerosumfree commutative semirings (Corollary 2.2.18).
(3) We establish a necessary and sufficient condition under which J -radical
and Js -radical coincide over semisimple semirings (Theorem 2.3.4), additively π regular semirings (Proposition 2.3.5, Theorem 2.3.6), antibounded left artinian
semirings (Theorem 2.3.10, Theorem 2.3.11) and left artinian left V-semirings
(Proposition 2.3.14, Theorem 2.3.16). These results partly answer [26, Problem
1].
(4) We give a complete description of semisimple semirings which are Js semisimple left (right) V-semirings (Theorem 2.4.1). Moreover, describe some
classes of such semirings that are Js -semisimple left (right) V-semirings, such
as simple semirings with an infinite element or congruence-simple left (right)
artinian left (right) subtractive semirings (Theorem 2.4.5), zeroic antibounded
semirings (Theorem 2.4.7). These results partly answer [1, Problem 1].
(5) The content of this chapter is based on the results in papers [53], [43],
[44] and [45].
17
Chapter 3
ON RADICAL CLASSES OF HEMIRINGS
In Chapter 3, we continue using the radical tool (here, for radical classes or
Kurosh-Amitsur radicals which are no specific radicals as chapter 2) to study the
structure of hemirings. We receive the concepts of J -radical and Js -radical of a
hemiring through the radical class of hemirings. We use to a concept of an accessible subhemiring (the similar concept of an accessible subring) to characterize
radical classes of hemirings, build a lover radical class of a class of hemirings
(in particular, build the lover radical class of a homomorphically closed class of
hemirings), establish a necessary and sufficient condition under which an upper
radical class of a regular class of hemirings is hereditary, prove an upper radical
class of a regular class of semirings is always hereditary. The main results of this
chapter are the results of papers [54] and [22].
3.1 Characterization of a radical class of hemirings by
accessible subhemirings
In this subsection, we recall concepts of a radical class, a radical operator,
a semisimple class and some results related to these concepts. Moreover, we
give an example on radical classes J and Js which have corresponding radicals
for J -radical and Js -radical of Chapter 2. Finally, we propose the concept of an
accessible subhemiring and give a characterization of a radical class of hemirings
by accessible subhemirings and hemiring homomorphisms.
Definition 3.1.1. ([48] or [15, p. 650]) A non empty subclass R of hemirings
of a universal class U is called a radical class of U if R satisfies the following two
conditions:
(1) R is homomorphically closed;
(2) For every hemiring S ∈ U \ R, there exists a subtractive ideal K ∈
I(S) \ {S} such that I(S/K) ∩ R = {0}.
Example 3.1.2. Let U be a universal class consisting of all hemirings.
(1) Set R := {R ∈ U | R is a ring}. Then, R is a radical class of U.
18
(2) A hemiring S is called right semiregular if for every s, t ∈ S there exist
x, y ∈ S such that
s + x + sx + ty = t + y + sy + tx.
Then, the subclass J := {S ∈ U | S is a right semiregular hemiring}, is a radical
class of U.
Next, we present a characterization of a radical class of hemirings.
Theorem 3.1.3. ([47, Theorem 3.2]) A subclass R of hemirings of a universal class U is a radical class of U if and only if R satisfies the following two
conditions:
(1) If S ∈ R then for every nonzero homomorphic image A of S there exists
a nonzero ideal B of A such that B ∈ R;
(2) If S ∈ U and for every nonzero homomorphic image A of S there exists
a nonzero ideal B of A such that B ∈ R then S ∈ R.
Example 3.1.8. Let U be a universal class consisting of all hemirings.
For every R ∈ U, denote ΣR := {R M ∈ |R M| | M is a simple left Rsemimodule}, Σ := ∪R∈U ΣR and F(Σ) := {R ∈ U | ΣR contains a faithful simple
left R-semimodule}. Then, according to [26, Proposition 3.1], F(Σ) is a regular
class. From [26, Proposition 3.5 and Theorem 3.2], the class
Js := {R ∈ U | Js (R) = R}
is a radical class of U and Js = UF(Σ).
Example 3.1.12. Let U be a universal class consisting of all hemirings.
(1) According to Example 3.1.2(2), the subclass J = {S ∈ U | S is a right
semiregular hemiring} is a radical class of U. Then, according to Theorem 3.1.11,
J-radical of a hemiring S ∈ U is the union of all right semiregular right ideals of
S and it is also a right semiregular ideal of S . Therefore, from Definition 2.1.4,
%J (S) = J(S).
(2) According to Example 3.1.8, the subclass Js = {R ∈ U | Js (R) = R} is a
radical class of U. Then, according to [26, Theorem 3.3], Js -radical of S ∈ U is
the intersection of all annihilators of simple left S -semimodules. And therefore,
%Js (S) = Js (S) for every S ∈ U.
The concept of an accessible subhemiring is a similar concept with an accessible subring in ring theory [11, p. 43].
19
Definition 3.1.15. A subhemiring S of a hemiring R is said to be accessible
if there exists a finite sequence of subhemirings S1 , S2 , ..., Sn of R such that
S = S1 C S2 C ... C Sn = R,
where Si C Si+1 (Si is an ideal of Si+1 ) but Si need not be an ideal of Si+2 or R.
Remark 3.1.16. According to Definition 3.1.15, if S is an ideal of hemiring R
then S is an accessible hemiring of R. However, the opposite is not true in general.
Indeed, consider the polynomial semiring R := B[x] over a Boolean semifield B.
Set S2 := {a2 x2 + ... + an xn | ai ∈ B, n ≥ 2} consisting of the zero polynomial and
all polynomials of degrees which are greater than or equal to 2 in R. Then, S2 is a
proper ideal of R. Set S1 := {a2 x2 +a4 x4 +a5 x5 +...+an xn | ai ∈ B, n ≥ 2} consisting
of the zero polynomial and all polynomials of degrees which are greater than or
equal to 2 but these polynomial do not contain terms of degree 3 in R. Then, S1
is a proper ideal of S2 and S1 is a subhemiring of R but S1 is not an ideal of R,
because x2 ∈ S1 and x ∈ R but x3 = x.x2 ∈
/ S1 . However, according to Definition
3.1.15, S1 is an accessible subhemiring of R.
According to Theorem 3.1.3, a radical class of hemirings is characterized
by ideals and hemiring homomorphisms. we characterize the radical class of
hemirings by accessible subhemirings and hemiring homomorphisms. This is a
similar result of [11, Theorem 3.1.9] in ring theory.
Theorem 3.1.17. A subclass R of hemirings of a universal class U is a
radical class of U if and only if R satisfies the following two conditions:
(1’) If R ∈ R then for every nonzero surjective homomorphisms f : R → S
there exists a nonzero accessible subhemiring I of S such that I ∈ R;
(2’) If R ∈ U and for every nonzero surjective homomorphisms f : R → S
there exists a nonzero accessible subhemiring I of S such that I ∈ R then R ∈ R.
3.2 On lower radical classes of hemirings classes
In this subsection, we build the lower radical class of a class of hemirings
by a similar method of Kurosh [34], and build the lower radical class of a class
of homomorphically closed hemirings by a similar method of Lee [40] in ring
theory.
The lover radical class of a class of hemirings is defined similar to the lover
radical class of a class of rings [11, p. 28]. Let A be a subclass of a universal
20
class U of hemirings, the intersection of all radical classes of U containing A is
a minimal radical class of U containing A, denoted by LA. The radical class LA
is called a lover radical class of U defined by A.
Next, we build the radical class of hemirings by a similar method of Kurosh
[34] in ring theory. Moreover, we prove that the radical class being built by this
method is a lower radical class of a class of hemirings.
Let A be an arbitrary subclass of a universal class U of hemirings. We define
the classes δλ (A) for each ordinal λ as follows. Define δ1 (A) to be a homomorphic
closure of A, i.e.,
δ1 (A) := {S ∈ U | S is a homomorphic image of a hemiring A ∈ A}.
Assume that δµ (A) has been defined for every ordinal µ < λ, we define δλ (A) as
follows:
δλ (A) := {S ∈ U | every nonzero homomorphic image of S has a nonzero ideal in
δµ (A) for some µ < λ}.
Finally, we define the class δ(A) := ∪δλ (A), where the union is taken over all
ordinals λ.
Theorem 3.2.2. Let A be an arbitrary subclass of a universal class U of
hemirings. Then, the subclass δ(A) is a radical class of U and δ(A) contains A.
Theorem 3.2.3. Let A be an arbitrary subclass of a universal class U of
hemirings. Then, δ(A) = LA.
Lee [40, Theorem 1] built the radical class from a homomorphically closed
class of rings. We end this subsection by building a radical class from a homomorphically closed class of hemirings by the similar method of Lee. Moreover,
we prove that the radical class being built by this method is a lover radical class
of a homomorphically closed class of hemirings.
Theorem 3.2.4. Let A be a homomorphically closed subclass of a universal
class U of hemirings. Then, the class
Y A := {S ∈ U | every nonzero homomorphic image of S has a nonzero
accessible subhemiring in A}
is a radical class of U and it contains A.
21
Corollary 3.2.5. If R is a radical class of a universal class U of hemirings
then Y R = R.
Corollary 3.2.6. If A is a homomorphically closed subclass of a universal
class U of hemirings then Y A is a lower radical class of A, i.e., Y A = LA.
Example 3.2.7. For a universal class U of all hemirings. Let A be a subclass
of U consisting of all nilpotent hemirings. Then, it is easy checked that A is a
homomorphically closed class, but A is no a radical class of U because the class
A does not have the inductive property in Theorem 3.1.4. Indeed, set Tn being
a hemiring of all upper triangular (positive rational) n × n matrices for n ≥ 2,
i.e.,
Tn := {(aij )n | aij ∈ Q+ , aij = 0 for i ≥ j}.
Then, Tn is a nilpotent hemiring of nilpotence degree n since Tnn = 0 but Tnn−1 6= 0.
Let us consider the direct sum R := ⊕∞
n=2 Tn of hemirings T2 , T3 , ..., Tn , .... In the
hemiring R there exists an ascending chain
T2 ⊆ T2 ⊕ T3 ⊆ ... ⊆ ⊕kn=2 Tn ⊆ ...
of ideals T2 , T2 ⊕ T3 , ..., ⊕kn=2 Tn , ... of R such that R is the union
k
R = ∪∞
k=2 (⊕n=2 Tn )
of the members of the chain, and each member of the chain is nilpotent, since
we have always
(⊕kn=2 Tn )k = 0.
However, the hemiring R is not nilpotent. Therefore, the subclass A consists all
nilpotent hemirings doing not satisfy the inductive property in Theorem 3.1.4.
According to Corollary 3.2.6, the class Y A is a minimal radical class of U
which contains all nilpotent hemirings.
Remark 3.2.8. Let R be a ring. Then, the intersection of all prime ideals of
R is called Baer radical [36]. According to [11, Example 2.2.2], the lower radical
class of a nilpotent hemirings class has a radical calculation being Baer radical.
Therefore, the radical class Y A of a class A contains all nilpotent hemirings
having a radical calculation corresponding with Baer radical in ring theory.
22
3.3 On hereditary radical classes of hemirings
In this subsection, we recall the such necessary and sufficient condition under
which a radical class of hemirings is hereditary, thence inferred radical classes
J and Js are hereditary. Next, we give a necessary and sufficient condition such
that an upper radical class of a regular class of hemirings is hereditary. Moreover,
we prove that the upper radical class of a regular class of semirings is always
hereditary, thence, the inferred Brown-McCoy radical class is hereditary.
Theorem 3.3.1. ([47, Theorem 6.2 and 6.4]) Let R be a radical class of a
universal class U of hemirings and %R be a corresponding radical operator. Then,
R is hereditary if and only if %R (I) = I ∩ %R (S) for each ideal I of any hemiring
S ∈ U.
According to Example 3.1.10 and Example 3.1.12, J and Js are corresponding radical operators for radical classes J and Js of a universal class U of all
hemirings. From Theorem 3.3.1, [21, Theorem 2] and Proposition 2.2.10, we
have the following corollary:
Corollary 3.3.2. For any universal class U of all hemirings, the radical
classes J and Js of U are hereditary.
Next, we consider the upper radical class of a regular class of hemirings but
it is not hereditary.
Example 3.3.3. ([47, Example 6.5]) For a universal class U of all hemirings,
P
let S ∈ U, denoted by S 2 := { ni=1 si ti | si , ti ∈ S}. Set M := {S ∈ U | S 2 = 0}.
According to Definition 3.1.6, M is a regular class in U. From Theorem 3.1.7,
the subclass
U M = {S ∈ U | if A 6= 0 is a homomorphic image of S then A ∈
/ M}
is an upper radical class of a regular class M. This upper radical class U M is not
hereditary. Indeed, consider the hemiring S := {0, a, e} ∈ U on which operations
of addition and multiplication are given by the following tables:
+ 0 a e
×
0 a
e
0
0
a
e
0
0
0
0
a
a
a
e
a
0
0
a
e
e
e
e
e
0 a
e
23
Since S has unit e so S ∈ U M. However, the ideal I = {0, a} of S satisfies I 2 = 0
so I ∈ M . This induces I ∈
/ U M. Therefore, the upper radical class U M is not
hereditary.
Next, we establish a necessary and sufficient condition to an upper radical
class of a regular class of hemirings is hereditary. Moreover, we prove that the
upper radical class of a regular class of semirings is always hereditary.
Theorem 3.3.4. Let M be a regular class of a universal class U of hemirings.
Then, the upper radical class U M is hereditary if and only if M satisfies the
following condition: If I is a nonzero ideal of S ∈ U and A ∈ M is a nonzero
homomorphic image of I , then there exists a nonzero homomorphic image B of
S such that B ∈ M.
Theorem 3.3.5. If M is a regular class of semirings in a universal class U,
then the upper radical class U M is hereditary.
According to Example 3.1.8(2) and Theorem 3.3.5, we have the following
consequence:
Corollary 3.3.6. For a universal class U of all hemirings, the Brown-McCoy
radical class of U is hereditary.
3.4 Conclusion of Chapter 3
(1) We introduce a concept of an accessible subhemiring (Definition 3.1.15)
and give a characterization of a radical class of hemirings by accessible subhemirings and homomorphisms (Theorem 3.1.17).
(2) We build a lower radical class of a class of hemirings by a similar method
of Kurosh in ring theory (Theorem 3.2.2 and Theorem 3.2.3). Using the concept
of an accessible subhemiring and a similar method of Lee in ring theory, we build
a lower radical class of a lass of homomorphically closed hemirings (Theorem
3.2.4 and Corollary 3.2.6).
(3) We prove that the radical classes J and Js are hereditary (Corollary
3.3.2). Establish a necessary and sufficient condition such that an upper radical
class of a regular class of hemirings is hereditary (Theorem 3.3.4). Prove an
upper radical class of a regular class of semirings is always hereditary (Theorem
3.3.5), thence, the Brown-McCoy radical class is hereditary (Corollary 3.3.6).
(4) The content of this chapter is results in papers [54] and [22].
24
CONCLUSION OF THE DISSERTATION
In this dissertation, we have obtained the following results:
(1) Using the concept of J -radical of hemirings, we give a complete description of J -semisimple additively π -regular semirings.
(2) We prove that there exists a simple left semimodule over an additively
idempotent hemiring and Js -radical coincides Nil radical over zerosumfree commutative semirings. From this result, we obtained a similar result of Snapper
on Jacobson radical of polynomial rings over commutative rings with unit for
zerosumfree commutative semirings. Moreover, we give a complete description
of Js -semisimple zerosumfree commutative semirings.
(3) Establish a necessary and sufficient condition under which J -radical and
Js -radical coincide over semisimple semirings, additively π -regular semirings, antibounded left artinian semirings and left artinian left V-semirings. From these
results, we partly answer [26, Problem 1]. Moreover, we describe some classes of
semirings which are Js -semisimple left (right) V-semirings. Thereby, we partly
answer [1, Problem 1].
(4) We propose a concept of an accessible subhemiring and give a characterization of a radical class of hemirings by accessible subhemirings and homomorphisms. Build a lower radical class of a class of hemirings by a similar method
of Kurosh in ring theory. Using the concept of an accessible subhemiring and a
similar method of Lee in ring theory, we build a lower radical class of a class of
homomorphically closed hemirings.
(5) Give such a necessary and sufficient condition that an upper radical class
of a regular class of hemirings is hereditary and we prove that an upper radical
class of a regular class of semirings is always hereditary. This result infers that
Brown-McCoy radical class of a universal class U of hemirings is hereditary.
25
THE LIST OF ARTICLES RELATED DIRECTLY TO
THE DISSERTATION
(1) Mai L. H. and Tuyen N. X. (2016), Some remarks on the Jacobson radical
types of semirings and related problems, Vietnam J. Math. (Online first).
(2) Inassaridze H., Mai L. H. and Tuyen N. X. (2014), On radical classes of
hemirings, Tbilisi Math. J., 7(1), pp. 69-74.
(3) Mai L. H. and Tuyen N. X. (2016), On Js -semisimple left (right) V-semirings,
J. Adv. Math. Stud., 9(3), pp. 437-443.
(4) Mai L. H. (2015), On radicals of left V-semirings, Hue Univ. J. Sci., 107(8),
pp. 87-94.
(5) Tuyen N. X. and Mai L. H. (2013), On a lower radical class and the corresponding semisimple class for semirings, Hue Univ. J. Sci., 82(4); pp. 207217.
THE RESULTS OF THE DISSERTATION WERE
REPORTED AND DISCUSSED ON
(1) The 8th Vietnamese Mathematical Conference, Telecommunications University, Nha Trang City, 08-2013.
(2) The Conference on groups, representations of groups and related problems,
Ho Chi Minh City University of Science, 11-2013.
(3) The 1st Central-Highland Conference on Mathematics, Quy Nhon University, 08-2015.
(4) The Conference on Algebra - Geometry - Topology, Buon Ma Thuot - Dak
Lak, 10-2016.
(5) The seminar of Department of Algebra-Geometry, Faculty of Mathematics,
College of Education, Hue University.